I have finally managed to teach Lucy to go after the object I throw(Whatever it is), grab it, carry it and bring it towards me. However I have a problem teaching her to let go of the object when I ask for it.

It gets more complicated: When I have a treat in my hand and she knows it, she brings the object to my feet and drops it asking for the treat. This works for me, but I would love it if she actually kept the object in her mouth and put it in my hand when I ask for it.

I've tried the "Give me" command much more than the actual fetch itself, but it seems to not work.

Without treats, she doesn't give up the object and just carries it away. Once I turn around and don't give her attention, she let's go.

With treats, she lets go of it almost when she's at my feet and close to her treat, but she never gives it to me as in holding it near my hand.

You've done well to teach her the idea of the fetch, now you just need to take a step back...

Teaching a dog to fetch is a series of behaviours:

1. Chase the item
2. Pick it up
3. Hold the item
4. Carry it
5. Come back to you with it
6. Give it back when you ask for it

So now, all I think you need to do is go back and train her to take it, hold it and give on its own (minus the retrieving part)! Some people need to start with food - just say take it as you offer a kibble. When she is eagerly opening her mouth to take the kibble, start with a toy. She should be eagerly opening her mouth to take the object after the food exercise. You can can praise and reward immediately or if she is doing well, you can delay for a second (which teaches her to hold it!). Then say give as you take it out of her mouth. Depending on the dog, it can be a really quick process or a rather long one! You may need to gently place your hand over her mouth while saying "hold it" (and if she knows the commands stay or wait, it might be a good time to say "Hold it...stay" so she doesn't move away!)

It sounds like she is only working for treats so remember, a dog should not know when a treat is coming - it is suppose to be variable! Also, once they understand the behaviour, you should no longer need to treat frequently (some would say you don't need to treat at all but I like to treat every once and while, even if my dog already knows the behaviour inside out!)

Good luck!

__________________
"Animals are reliable, many full of love, true in their affections, predictable in their actions, grateful and loyal. Difficult standards for people to live up to."